qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,584,927 | <p>If I have two isomorphic groups, can I write <span class="math-container">$A \xrightarrow{\sim} B$</span> rather than <span class="math-container">$A \cong B$</span> to mean "A is isomorphic to B", or is the arrow notation only used if I have a map <span class="math-container">$\varphi : A \xrightarrow{\sim} B$</spa... | Shaun | 104,041 | <p>Both notations are accepted. </p>
<p>If you're using either of them, though, it is good practise to state what it means; it is usually left as implicit from the context.</p>
<p>Having said that, it should be noted that "<span class="math-container">$A\stackrel{\sim}{\to}B$</span>" usually refers to a specific isom... |
2,202,048 | <p>There's a box with $N$ balls in it. One of them is red and the others white. What's the probability of getting the red ball at the $k$th try ( if you're not putting them back in?) where $k = 1,2,3,\ldots,N$</p>
| Fimpellizzeri | 173,410 | <p>You can compute the probability to be $1/N$, independently of $k$, directly by noticing that for the first $k-1$ draws you mustn't draw the red ball.</p>
<p>That said, another way to see this is that drawing a single ball from the box in succession until it is empty can be represented by an ordering of the balls (w... |
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | antkam | 546,005 | <p>All the answers are basically saying the same thing, so I will simply try to say it in the most "graph-theoretic" way. :)</p>
<hr>
<p>Construct an <span class="math-container">$n$</span>-node tree <span class="math-container">$G$</span> with the node degrees being <span class="math-container">$a_1, a_2, ..., a_n$... |
2,912,401 | <p>I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy sequence. </p>
<p>I know that a sequence of real numbers is <em>not</em> Cauchy if there exists an $\epsilon>0$ such ... | Mark Bennet | 2,906 | <p>You only need such an $\epsilon$ to exist, so you can choose a convenient value.</p>
<p>Then you need to show that there are gaps bigger than $\epsilon$ between elements of the sequence as far a=out as you care to go - that gaps as big as $\epsilon$ don't fade out and disappear in the tail of the sequence.</p>
<p>... |
3,122,612 | <p>I am doing a presentation on nonlinear optics and I ran into a paper that uses a series to describe a wave equation of a system. The paper will be linked below and the series is:</p>
<p><span class="math-container">$\sum_{n=1}^{\infty}\frac{(-i)^{n-1}}{n!}k^n$</span></p>
<p>I mostly am looking for what i should ca... | JV.Stalker | 416,274 | <p>Think the name of the function is "Exponential generating function (EGF)" see Wikipedia. </p>
<p><span class="math-container">$\sum\limits_{n=1}^\infty ik_n \frac{(-i)^n}{n!}=EGF(ik_n,-i)-ik(0)$</span></p>
|
1,025,588 | <p>I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem:</p>
<p>If $a, b, c \gt 0$ prove that $$ a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a.$$</p>
<p>I am going crazy over this! A hint or pro... | Community | -1 | <p>$$\color{blue}{a^2(a-b)+b^2(b-c)+c^2(c-a)=(a^2-c^2)(a-b)+(b^2-c^2)(b-c)}$$</p>
<p>$$\color{red}{a\geq b \geq c \implies (a^2-c^2)(a-b)+(b^2-c^2)(b-c) \geq0}$$ </p>
|
2,174,454 | <p>My teacher substitutes for this $\sum_{x=0}^{y} ({{y!} \over {x! (y-x)!} })$ by $2^y$, so I tried to use the (Mathematical induction) to prove it (My teacher did not ask me to do that, however I want to do it only to make sure for this statement)</p>
<p>My attempt:</p>
<ul>
<li>when $y=1$ :</li>
</ul>
<p>L.H.S.
$... | MAN-MADE | 422,850 | <p>You can verify it easily. The terms in LHS are $\frac{y!}{x!(y-x)!}$ which means the way you can choose $x$ objects form $y$ objects. The sum over $x$ will give you total ways to select a object from $y$ objects which is $2^y$ since it is about select or not select i.e. 2 ways, i.e. in all total $2^y$ ways.</p>
<p>... |
14,527 | <p>The <a href="https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_algorithm" rel="nofollow noreferrer">Robinson-Schensted correspondence</a> is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.</p>
<p>Some simple operations on tableaux cor... | darij grinberg | 2,530 | <p>Edit: This doesn't answer the question; see comments.</p>
<p>Knuth, TAoCP3, p. 76, exercise 5 in section 5.1.4:</p>
<p>Let $P$ be the tableau corresponding to the permutation $a_1a_2...a_n$; use exercise 4 to prove that $P^T$ is the tableau corresponding to $a_n...a_2a_1$.</p>
|
14,527 | <p>The <a href="https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_algorithm" rel="nofollow noreferrer">Robinson-Schensted correspondence</a> is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.</p>
<p>Some simple operations on tableaux cor... | Richard Stanley | 2,807 | <p>It doesn't seem natural to the theory of RSK to transpose both tableaux. When we apply
RSK to <span class="math-container">$a_n\cdots a_1$</span> then we get the pair <span class="math-container">$(P^t,\mathrm{evac}(Q)^t)$</span>, where evac is
Schutzenberger's evacuation operation. See for instance <em>Enumerative ... |
183,058 | <p>I'm having difficulty producing a proper animated gif (for inclusion in a PowerPoint presentation). The code I have is this:</p>
<pre><code> HSBhsl[h_, s_, l_] :=
Module[{H, S, B}, H = h; B = l + s (1 - Abs[2 l - 1])/2;
S = (2 B - l)/B; {H, S, B}]
TestCube[\[Theta]_] :=
Graphics3D[Rotate[{EdgeForm[], Hue[HSB... | rafalc | 51,110 | <p>I have investigated the problem and it seems that this is indeed a limitation of <code>Export</code>. We will add support for <code>"TransitionEffect"</code> option in <code>Export</code> in the next release of Mathematica.</p>
<p>In the meantime, as a workaround, you can use a low-level function from GIFTools pacl... |
4,322,897 | <p>Let <span class="math-container">$\{y_n\}_{n=1}^\infty$</span> be a sequence in <span class="math-container">$\Bbb{R}^n$</span> and <span class="math-container">$z\in\Bbb{R}^n$</span> be given. If <span class="math-container">$\{\|z-y_n\|\}_{n=1}^\infty$</span> is a convergent sequence, what can we say about <span c... | LGu | 987,256 | <p>First let the sequence WLOG be unbounded from above, i.e. for each <span class="math-container">$z\in R$</span>, <span class="math-container">$\exists k$</span> such that <span class="math-container">$y_k>z$</span>. Now let <span class="math-container">$\epsilon>0$</span> be given, if <span class="math-contain... |
2,489,429 | <p>$\int \frac{x}{x+1}\,dx$ so I set substitution to $x+1=t$ then I differentiate to get $dx=dt$ and then proceed to get $\int 1\,dt - \int \frac{1\,dt}{t}$ to get $t-\ln(|t|)+C$ but when I replace back the substitution I get $x+1$ in the final answer instead of just the $x$ , so my question is why was it a mistake to ... | Cornman | 439,383 | <p>Alternative:</p>
<p>$\int \frac{x}{x+1}\, dx=\int \frac{x+1-1}{x+1}\, dx=\int 1-\frac{1}{x+1}\, dx= x-\ln|x+1|+C$</p>
<p>Without substitution.</p>
|
153,999 | <p>How to solve: $|\sqrt{x-1}-2| + |\sqrt{x-1}-3|=1$.</p>
<p>I would like to know how to solve an absolute value equation when there is a square root sign inside.</p>
| Steven Alexis Gregory | 75,410 | <p>Theorem: Let $X,P,Q \in \mathbb R^n$. Then $X \in \overline{PQ} \iff d(P,X) + d(X,Q) = d(P,Q)$</p>
<p>On the set of real numbers, the distance between numbers $x$ and $y$ is $|x-y|$. So we have this.</p>
<p>Corollary: Let $x,p,q \in \mathbb R$ with $p < q$. Then $p \le x \le q \iff |x-p| + |q-x| = q-p$</p>
<p>... |
3,114,663 | <p>Let there be <span class="math-container">$m$</span> indistinguishable balls, <span class="math-container">$k$</span> bins, <span class="math-container">$C$</span> capacity.
Let <span class="math-container">$X_j$</span> denote the total balls in bin <span class="math-container">$j$</span>.
I've seen ways to calcula... | parsiad | 64,601 | <p>Here's some a Monte Carlo simulation in Python for the problem. Disclaimer: it's not efficient.</p>
<pre><code>import numpy as np
import sys
def simulate(num_balls, num_bins, capacity, num_sims):
assert(capacity * num_bins >= num_balls)
results = np.empty((num_sims,))
for i in xrange(num_sims):
... |
1,928,892 | <p>I'm solving a graduate entrance examination problem.
We are required to establish the inequality using the following result:</p>
<p>for $x,y > 0$, $\frac{x}{y} + \frac{y}{x} > 2$ (1), which is easy to prove as it is equivalent to $(x - y)^2 > 0$.</p>
<p>But when it comes to an inequality combining $x, y,... | Zuo | 399,908 | <p>Proof:</p>
<p>If $x,y,z>0$,then:$$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$$$\geq3(xy+yz+zx)$$</p>
<p>So:$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}$$$$=\frac{x^2}{xy+xz}+\frac{y^2}{xy+yz}+\frac{z^2}{yz+zx}$$$$\geq\frac{(x+y+z)^2}{2(xy+yz+zx)}\geq\frac3{2}$$</p>
|
2,736,596 | <p>I want to proof this equation below.
$$
f^{-1}(\bigcup B_{j})=\bigcup(f^{-1}B_{j})
$$
I know it's true but I don't have idea how to proof and explain.
Please help.</p>
| drhab | 75,923 | <p>Essential is: $$x\in f^{-1}(A)\iff f(x)\in A$$</p>
<p>So the following statements are equivalent:</p>
<ul>
<li>$x\in f^{-1}(\bigcup_{j\in J}B_j)$</li>
<li>$f(x)\in\bigcup_{j\in J}B_j$</li>
<li>$f(x)\in B_{j_0}$ for some $j_0\in J$</li>
<li>$x\in f^{-1}(B_{j_0})$ for some $j_0\in J$</li>
<li>$x\in\bigcup_{j\in J}f^... |
2,487,044 | <p>Why are integral elements defined in terms of monic polynomials? Why do we wish to split the cases between non monic polynomials and monic polynomials? I.e, algebraic elements and integral elements. Motivation? </p>
| reuns | 276,986 | <p>Try with $\sqrt{2}$ then the ring $\mathbb{Z}[\sqrt{2}]$ is just $\{ a+b\sqrt{2}, (a,b) \in \mathbb{Z}\}$. With $\frac{\sqrt{2}}{2}$ then the ring $\mathbb{Z}[\frac{\sqrt{2}}{2}]$ is $\{\frac{a+b\sqrt{2}}{2^k}, (a,b,k) \in \mathbb{Z}\}$. The difference is that $\frac{\sqrt{2}}{2}$ is not the root of any monic polyno... |
2,990,400 | <p>I need to find all the continuous functions <span class="math-container">$f$</span> such that
<span class="math-container">$$
[f(t)]^2=\int_0^t f(s) ds
$$</span>
<strong>Attempt:</strong></p>
<p>Since
<span class="math-container">$$
f(t)\leq [f(t)]^2+c, \, \, \forall c \geq \frac14$$</span>
we have</p>
<p><span ... | TurlocTheRed | 397,318 | <p>You can use the Lemma use to prove the Fundamental Theorem of Calculus:</p>
<p><span class="math-container">$$\frac{d}{dx}\int_a^xf(t)dt=f(x)$$</span></p>
<p>Here we have <span class="math-container">$[f(t)]^2=\int_0^tf(s)ds.$</span></p>
<p>Take the <span class="math-container">$t$</span> derivative of both sides... |
2,238,165 | <p>Can anyone help with this limit?</p>
<p>\begin{equation*}
\lim_{x \rightarrow 4}
\frac{16\sqrt{x-\sqrt{x}}-3\sqrt{2}x-4\sqrt{2}}{16(x-4)^2}
\end{equation*}</p>
<p>I've tried a variable change of \begin{equation*} y=\sqrt{x} \end{equation*} but this didn't help.</p>
| Especially Lime | 341,019 | <p>You can almost calculate it without any cases, but there is a slight extra complication from the fact that the first digit can't be $0$.</p>
<p>If you just wanted to know how many strings of $N$ digits with exactly two $5$s and three $7$s, then you can first choose two places from $N$ for the $5$s, then three place... |
3,245,854 | <p>Hi I am trying to solve an integral problem that involves trig substitution. First I tried completing the square, which gave me <span class="math-container">$1/\sqrt{(x+3)^2+2^2}$</span>. I know I am supposed to use <span class="math-container">$x = \arctan(\theta)$</span>. Does that mean it should be: </p>
<p><spa... | John Wayland Bales | 246,513 | <p><span class="math-container">$$\int \frac{1}{\sqrt{x^2+6x+13}}\thinspace dx=\int\frac{1}{\sqrt{(x+3)^2+2^2}}\thinspace dx$$</span>
<a href="https://i.stack.imgur.com/lv0K4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lv0K4.png" alt="Trig substitution"></a></p>
<p>So <span class="math-container... |
131,583 | <p>My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any <strong>algorithm</strong>) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, Hodg... | Victor Lehenkyi | 39,966 | <p>Basic paper is
Journal of Mathematical Physics , Vol. 12. New York: American Institute of Physics. (1971): 653-666 (so called "Harrison-Estabrook procedure.")
This procedure is implemented in package "Liesymm" for Maple.</p>
|
551,471 | <p>In a proof I am reading, the author considers the situation where $G$ is a reductive algebraic group (variety) over the complex numbers $\mathbb C$ and $N\trianglelefteq G$ is a closed, normal subgroup of $G$ which is also finite. The author concludes that <em>"because $G$ is reductive"</em>, $N$ must be contained i... | user466484 | 466,484 | <p>Actually, any finite normal subgroup H of a connected algebraic group G is central. (Consider the action GxH->H by conjugation. The orbits are connected.) </p>
|
2,077,053 | <p>Please check my proof</p>
<p>If $H\triangleleft G$ then $gHg^{-1}=H $ for every $g\in G $</p>
<p>If $K\triangleleft G$ then $gKg^{-1}=K$ for every $g\in G$</p>
<p>Then $H\cap K = gHg^{-1}gKg^{-1}=gHKg^{-1}$</p>
<p>Then $H\cap K\triangleleft G$</p>
| Zelos Malum | 197,853 | <p>You proof is flawed as
$$H\cap K \subseteq H\subseteq HK$$
where equality does not need to hold so you cannot say that.</p>
<p>A better method is this to show that for $x\in H\cap K$ we have $gxg^{-1}\in H\cap K$ for all $g$.</p>
<p>This is trivial as $x\in H$ gives that $gxg^{-1}\in H$ and $x\in K$ gives that $gx... |
2,254,694 | <p>I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:</p>
<p>Attempt 1:</p>
<p>Write out the terms of $b_n$:</p>
<p>$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a_{3}-\frac{a_{2}}{2}$$
$$b_3=a_{4}-\frac{a_{3}}{2}$$
$$\cdots$$
$$... | Michał Miśkiewicz | 350,803 | <p>Let $c_n = 2^n a_n$. The assumption reads
$$ \frac{c_{n+1}-c_n}{2^{n+1}-2^n} \to 0. $$
The sequence $2^n$ tends monotonically to $+\infty$, hence by the <a href="https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="nofollow noreferrer">Stolz–Cesàro theorem</a>
$$ a_n = \frac{c_n}{2^n} \to 0. $$</p... |
1,517,245 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Arrangements of $1,2,..,81$ in an $9 \times 9$ matrix such that each row and each column is increasing. How many possible $X$ ?</p>
<p><a href="https://i.stack.imgur.com/Te1TB.png" rel="nofollow noreferrer"><img src="https://i.stack.... | Martin Brandenburg | 1,650 | <p>They are sometimes also called <strong>pointed monoids</strong>. See <a href="https://arxiv.org/abs/1909.00297v1" rel="nofollow noreferrer">arXiv:1909.00297</a> for instance.</p>
|
2,008,243 | <p>How to find the exact value of the following limit?</p>
<p>$\lim_\limits{x \to 0}\frac{e^{x\cos{x^2}}-e^{x}}{x^5}$</p>
| Nicolas FRANCOIS | 288,125 | <p>$$\cos x^2=1-\frac{x^4}{2}+o(x^4)$$
so
$$e^{x\cos x^2} = e^{x-\frac{x^5}{2}+o(x^5)} = e^xe^{-\frac{x^5}{2}+o(x^5)}$$
therefore
$$\frac{e^{x\cos x^2}-e^x}{x^5} = e^x\frac{e^{-\frac{x^5}{2}+o(x^5)}-1}{x^5} \sim e^x\frac{-\frac{x^5}{2}}{x^5} \xrightarrow[x\to0]{}-\frac{1}{2}$$
using $e^u-1\sim u$ when $u\to0$.</p>
|
5,581 | <p>(Wade's rewording)</p>
<p>There is a phrase that contains $4$ words and $22$ total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and $12$ letters appear once each. What is this phrase?</p>
<blockquote>
<p>@@$$#... | David E Speyer | 448 | <p>Captain Kirk was wondering why his journal was not more excited to hear about his exploits. He said</p>
<p>LOG, YOU SEEM FAINTHEARTED!</p>
<hr>
<p>More seriously, I dumped the letters EEEETTAAOOINSHRDLUGFMY into a multiword anagrammer, and it immediately found more than 24000 answers, of which this was one of the... |
578,086 | <p>Hey I've been having problem finding the limit of the sequence below</p>
<p>$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}$</p>
<p>Thanks!</p>
| gangrene | 103,440 | <p>The solution I know involves the <em>p-norms</em> and the <em>Gamma function</em>: in fact let $f: [0, \infty[ \to \mathbb{R}$ be defined by $f(x)=xe^{-x}$. Since $f$ is positive, continuous, and $0=f(0)=f(\infty)$, we have $\|f\|_\infty = \max f([0,\infty])$; now $f'(x)=(1-x)e^{-x}$ is $0$ for $x=1$, so that $\|f\|... |
1,279,947 | <p>In physics, 1m is defined 'the length which light travels in certain time'.
And i've thought 2m has twice many elements than 1m.
But today i've realized intervals (0,1) and (0,2) are equipotent.
I was shocked because i've thought length and cardinality are similar concepts.
Cardinality is clearly defined in set theo... | 5xum | 112,884 | <p>The best "approximation" of length in mathematics is "measure theory". This is an extensive field of mathematics that deals with "sizes" (called "measures") of sets. </p>
<p>You can define a measure on $\mathbb R$ on which the measure of an interval $(a,b)$ is equal $b-a$. This is called the Lebesgue measure, and i... |
28,586 | <p>Imagine I have two lists:</p>
<pre><code>List1 = {0, 1, 0.6, 0.5, 1.2, 0.4};
List2 = {"a", "b", "c", "d", "e", "f"};
</code></pre>
<p>How can I use <code>Pick</code> to return a third list consisting of items in List2 that have a array position corresponding to an array position with a value $\leq N$ in the first ... | chyanog | 2,090 | <p>You can also use <code>UnitStep</code>, it has the Listable attribute, listable operations will be faster than other alternatives.</p>
<pre><code>List1 = {0, 1, 0.6, 0.5, 1.2, 0.4};
List2 = {"a", "b", "c", "d", "e", "f"};
Pick[List2, UnitStep[0.5 - List1], 1]
(*{"a", "d", "f"}*)
</code></pre>
|
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| robjohn | 13,854 | <p>Compute the generating function of the harmonic numbers:
$$
\begin{align}
\sum_{n=1}^\infty H_nx^n
&=\sum_{n=1}^\infty\sum_{k=1}^n\frac{x^n}{k}\\
&=\sum_{k=1}^\infty\sum_{n=k}^\infty\frac{x^n}{k}\\
&=\sum_{k=1}^\infty\sum_{n=0}^\infty\frac{x^{n+k}}{k}\\
&=-\frac{\log(1-x)}{1-x}\tag{1}
\end{align}
$$
... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| r9m | 129,017 | <p>Starting with $\displaystyle \dfrac{H_n}{n} = \sum_{k=1}^{\infty} \dfrac{1}{k(k+n)}$ we have,
\begin{align*}\sum_{n=1}^{\infty} \dfrac{H_n^2}{n^2} &= \sum_{n=1}^{\infty} \left(\sum_{k=1}^{\infty}\dfrac{1}{k(k+n)}\right)^2\\&= \sum_{n=1}^{\infty}\sum_{k,j=1}^{\infty} \dfrac{1}{jk(j+n)(k+n)} \\&= \sum_{n=1... |
3,081,552 | <p>Is this <code>∅ ⊈ { ∅, 1, 2 }</code> true or false ? </p>
<p>Also, I am confuse since this <code>{ ∅, 1, 2 }</code> has already contain a <code>∅</code>, does it still contain another <code>∅</code> meaning like : <code>{ ∅, ∅, 1, 2 }</code> ? </p>
<p>Is <code>∅ ∈ { ∅, 1, 2 }</code> true ? & <code>{∅} ∈ { ∅,... | Anik Bhowmick | 354,636 | <p>No, not like that. Since null set is subset of all sets, the question in your title is valid. At least I think so.</p>
|
2,390,804 | <p>How does one compute the following integral?
$$\int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx$$</p>
<hr>
<p>I have tried extending $x$ to the complex plane then evaluating the following contour integral
$$\oint_C \frac{\sqrt{x}e^{ix}}{1+x^2} dx$$
with the contour $C$ running along the whole real axis and then... | Random Variable | 16,033 | <p>Using the Laplace transform of the sine function, we get <span class="math-container">$$ \begin{align} \int_{0}^{\infty} \frac{\sqrt{x}\sin x}{1+x^{2}} \, \mathrm dx &= \int_{0}^{\infty} \sqrt{x} \sin(x) \int_{0}^{\infty} \sin(t) e^{-xt} \, \mathrm dt \, \mathrm dx \\ &= \int_{0}^{\infty} \sin(t) \int_{0}... |
63,534 | <p>I found in an article <a href="http://dx.doi.org/10.1103/PhysRev.105.776" rel="nofollow">"Imperfect Bose Gas with Hard-Sphere Interaction"</a>, <em>Phys. Rev.</em> 105, 776–784 (1957) the following integral, but I don't know how to solve it. Any hints?</p>
<p>$$\int_0^\infty {\int_0^\infty {\mathrm dp\mathrm dq\fr... | Andrew | 11,265 | <p>First of all one can note that the integral converges and is a differentiable function of parameters for $4vw>u^2$. With the change of variables $p\to p' u^{1/2}$, $q\to q' u^{1/2}$ the general case be reduced to $u=1$. Now denoting the lhs
$$
f(v,w)=\int_0^\infty {\int_0^\infty {\mathrm dp\mathrm dq\fr... |
13,467 | <p>I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2;</p>
<p><strong>Type I</strong></p>
<blockquote>
<p>Consider an experiment of rolling two dice:</p>
<p>Sample space $$ S = \{(1,1),(1,2),(1,3), \cdots, (6,6... | Andreas Blass | 326 | <p>Since they can do Type-1 problems, I'd begin by trying to make your Type-2 problem look as similar to Type 1 as possible, despite the fact that, in the Type-2 problem, we don't actually know the size of the sample space. So I'd temporarily assume some specific size, say $6$ (to make the arithmetic easy). So we have... |
3,879,366 | <p>This might or might not be author specific, but in Algebraic Statistics by Seth Sullivant the author uses a notation of <span class="math-container">$S_{K, K}$</span> for example. So far I have not been able to find out what it means. Here is an <a href="https://i.stack.imgur.com/oEJHw.png" rel="nofollow noreferrer"... | Zac | 521,980 | <p>I haven't solved it yet but figured that the function is reasonably inspectable via plotting in this case. Some simple python code allows us to visually identify where the minimum should be, although from the answers so far no analytical solution appears forthcoming:</p>
<pre><code>import numpy as np
import matplotl... |
4,141,812 | <p>Considering the vector space <span class="math-container">$\mathbb{R}^3$</span>, find all values of <span class="math-container">$k$</span> such that
<span class="math-container">\begin{align*}
\begin{bmatrix}
2k^2 \\
-3k \\
1
\end{bmatrix} \in \text{ span }
\left\{
... | Ritam_Dasgupta | 925,091 | <p>Yes, the vertical asymptote would occur at some <span class="math-container">$x=a$</span> such that: <span class="math-container">$$e^a=-a$$</span>
I do not think there is any elementary way to find this <span class="math-container">$a$</span>, but you could plot it and find out.</p>
|
692,383 | <p>Let $p, q$ be prime numbers which may or may not be distinct.
Let $\mathbb{Q}_p$ be the field of $p$-adic numbers.
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers.
We define similarly $\mathbb{Q}_q$ and $\mathbb{Z}_q$.</p>
<p>Let $A = \mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$.
Let $\lambda\colon \mathbb{Z}... | Makoto Kato | 28,422 | <p>All rings will be assumed to be commutative with unity.</p>
<p><strong>Lemma 1</strong>
Let $A$ be a ring, $M, N$ $A$-modules.
Suppose $N$ is flat over $A$.
Let $a$ be an element of $A$.
Suppose the map $f\colon M \rightarrow M$ defined by $f(x) = ax$ is injective.
Then the map $g\colon M\otimes_A N \rightarrow M\o... |
712 | <p>I read <a href="https://math.stackexchange.com/questions/625/why-is-the-derivative-of-a-circles-area-its-perimeter-and-similarly-for-spheres">this question</a> the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle. </p>
... | user76995 | 76,995 | <p>The area of a circle can be defined as a summation of infinite concentric circles which are infinitely small, each of radius $r$ perimeter $2\pi r$ where $r$ is a variable that changes from $0$ to $R$, where $R$ denotes the radius of the circle we defined.</p>
<p>So, obviously when the area of the circle is differe... |
47,728 | <p>Function <code>CreateTemporary</code> can generate a unique file name like</p>
<pre><code>file = CreateTemporary[]
(* "C:\Documents and Settings\kkouptsov\Local \Settings\Temp\m-263a0380-3a03-49ac-95cb-d21390b2d3fd" *)
</code></pre>
<p>How do we generate a unique file name such as above example? The reason I don't... | Sascha | 4,597 | <p>Version 10.0 introduced <code>CreateUUID</code></p>
<pre><code>CreateUUID[]
(* "1aaeab6f-ad51-48d2-a939-366da47374a3" *)
</code></pre>
<p>The <a href="http://reference.wolfram.com/language/ref/CreateUUID.html" rel="nofollow noreferrer">documentation</a> says that</p>
<blockquote>
<ul>
<li><p>CreateUUID[] g... |
941,596 | <p>Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?</p>
| Ri-Li | 152,715 | <p>There is no known distribution function but you can create by your own.
Order the biased coin $1,2 \ldots ,M$. Now in the binomial coefficient we have $\binom Mk \times p^k \times q^{M-k}$ . Here for a fixed $k$ it will be $ (p_{i_1}p_{i_2}p_{i_3} \ldots p_{i_k})(q_{i_1}q_{i_2}q_{i_3} \ldots q_{i_k})$ product will r... |
83,236 | <p>I need a <code>compare-operation</code> for two lists of same length (usually > 100000) which does the following:</p>
<blockquote>
<p>{2,3,5,4,1,8,7} <code>compare-operation</code> {1,4,6,3,2,8,8} = {2,4,6,4,2,8,8}</p>
</blockquote>
<p>The resulting list has at each position the greater (or equal element in case... | LLlAMnYP | 26,956 | <pre><code>MapThread[Max, {{2, 3, 5, 4, 1, 8, 7} , {1, 4, 6, 3, 2, 8, 8}}]
</code></pre>
<blockquote>
<p><code>{2, 4, 6, 4, 2, 8, 8}</code></p>
</blockquote>
<pre><code>Thread[Unevaluated[Max[{2, 3, 5, 4, 1, 8, 7} , {1, 4, 6, 3, 2, 8, 8}]]]
</code></pre>
<p>also works. <code>Unevaluated</code> is necessary, otherw... |
83,236 | <p>I need a <code>compare-operation</code> for two lists of same length (usually > 100000) which does the following:</p>
<blockquote>
<p>{2,3,5,4,1,8,7} <code>compare-operation</code> {1,4,6,3,2,8,8} = {2,4,6,4,2,8,8}</p>
</blockquote>
<p>The resulting list has at each position the greater (or equal element in case... | Carl Woll | 45,431 | <p>For threading the maximum of a list (or array) with a number, you'll be hard pressed to beat the following:</p>
<pre><code>ThreadedMax[l1_List, l2_?NumericQ] := Clip[l1, {l2, Infinity}]
</code></pre>
<p>For example:</p>
<pre><code>l1 = RandomReal[1, 10^8];
ThreadedMax[l1, .5]; //AbsoluteTiming
</code></pre>
<bl... |
4,340,560 | <p>I have to teach limits to infinity of real functions of one variable.
I would like to start my course with a beautiful example, not simply a basic function like <span class="math-container">$1/x.$</span> For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the bas... | Exponent BOT | 404,974 | <p>Explaining the failure of Zeno's Paradox could be a cool activity. Motion can be broken down into so many very small parts, namely steps of length <span class="math-container">$1/2^i$</span>. Show the fact that <span class="math-container">$\lim_{n \to \infty} \sum_{i=1}^n 1/(2_i) =1$</span>. The only prerequisite o... |
3,009,351 | <p>Consider the vectors <span class="math-container">$q_1=(1,1,1)$</span> and <span class="math-container">$q_3=(1,1,-2)$</span>. I need to find a third vector <span class="math-container">$q_2$</span> such that <span class="math-container">$\{q_1,q_2,q_3\}$</span> is a arthogonal basis for <span class="math-container... | Siong Thye Goh | 306,553 | <p><span class="math-container">$$q_2=v-\frac{\langle v|q_1\rangle}{\langle q_1|q_1\rangle} q_1-\frac{\langle v|q_3\rangle}{\langle q_3|q_3\rangle} q_3$$</span></p>
|
3,387,843 | <p>The thing I want to know is that why the term <span class="math-container">$``vector"$</span>.</p>
<p>I had learned that vector is something that has magnitude and direction, and also the the elements of <span class="math-container">$\mathbb{R}^n$</span> i.e. <span class="math-container">$n$</span> tuples can be vi... | user | 505,767 | <p>The ordinary geometric vectors in <span class="math-container">$\mathbb R^2$</span> or in <span class="math-container">$\mathbb R^3$</span> are a special kind of a more general structure named <a href="https://en.m.wikipedia.org/wiki/Vector_space" rel="nofollow noreferrer">vector space</a> which obeys to the same ru... |
3,154,947 | <p>Hello MathStackExchange Community, </p>
<p>Confused about how to change the notation-formula for permutations when you have more complex processes. i.e. multiple independent action choices of different types of objects and orientation limitations. </p>
<ul>
<li><em>The generic formula is : <span class="math-contai... | B. Goddard | 362,009 | <p>I think this is a Green's theorem problem. The loop is traced as <span class="math-container">$t$</span> goes from <span class="math-container">$-4$</span> to <span class="math-container">$4$</span> (but clockwise.) So the area is</p>
<p><span class="math-container">$$-\frac{1}{2} \int_{-4}^4 x \; dy - y \; dx $$... |
475,267 | <p>Suppose $f(0) =0 $ and $0<f''(x)<\infty (\forall$ $x>0)$, then $\frac{f(x)}{x}$ strictly increases as $x$ increases. </p>
<p>I have shown that $f'(x)-\frac{f(x)}{x} = \frac{1}{2}xf''(c)$, for some $c\in (0,x)$. How do I proceed from here? </p>
| Pete L. Clark | 299 | <p>The hypothesis <span class="math-container">$f'' \geq 0$</span> means that your function is <strong>convex</strong>. There are many standard inequalities involving convex functions. This is a special case of one of them, which I call the <strong>two secant inequality</strong>:</p>
<blockquote>
<p>If <span class="m... |
2,307,399 | <blockquote>
<p>How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor $$
For a given $n$. </p>
</blockquote>
<p>For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets complicated . Is there a way to calculate this summation?</p>
| infinitylord | 178,643 | <p><strong>disclaimer:</strong> This is not a complete answer, just an approximation I came up with that was too long to comment.</p>
<p>If we truncate $e $ at $3$ decimal places (so it is 2.718), we can get a decent approximation:</p>
<p>$$\sum_{i=1}^n \lfloor 2.718i \rfloor \approx \frac32 (n(n+1) - \lfloor \frac{n... |
2,307,399 | <blockquote>
<p>How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor $$
For a given $n$. </p>
</blockquote>
<p>For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets complicated . Is there a way to calculate this summation?</p>
| Matthew Conroy | 2,937 | <p>Let's approximate $e$ by a rational number $A$ such that we have a positive integer $k$ such that $10^kA$ is an integer (e.g., $A=2.71$ with $k=2$).</p>
<p>We can then use the fact that $\lfloor (m\cdot10^k+i)A \rfloor = m\cdot 10^k A+\lfloor Ai \rfloor$ for any positive integer $m$.</p>
<p>Then, for a positive in... |
2,297,542 | <blockquote>
<p>Given integers $a,b,c,d$ with $d\not\equiv0\bmod5$ and $m$ an integer for which $am^3+b m^2+cm+d\equiv0\bmod5$, prove that there exists an integer $n$ for which $dn^3+cn^2+bn+a\equiv0\bmod5$.</p>
</blockquote>
| Landon Carter | 136,523 | <p>Let $Y\sim Exp(1)$ then $-Y$ has the desired cdf.</p>
|
3,469,392 | <p>Let <span class="math-container">$a_n$</span> be a recursive sequence where <span class="math-container">$a_1=42$</span> and for all <span class="math-container">$n$</span>:</p>
<p><span class="math-container">$$a_{n+1}=a_n+\frac{(-1)^n(1+n!)}{2^nn!}$$</span></p>
<p>After plugging some values of <span class="math-... | Robert Z | 299,698 | <p>We have that
<span class="math-container">$$a_{N+1}=a_1+\sum_{n=1}^N\frac{(-1)^n(1+n!)}{2^nn!}=42+
\sum_{n=1}^N\frac{(-1/2)^n}{n!}+\sum_{n=1}^N\left(-\frac{1}{2}\right)^n$$</span>
and as <span class="math-container">$N\to +\infty$</span> both series can be easily evaluated: the first one is related to the <a href="h... |
432,223 | <p>I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do.</p>
<p>Help appreciated.</p>
<p>Thanks.</p>
| Glad Galaxy | 241,926 | <p>I think here for factoring the polynomial $x^6+5x^3+8$ we must go ahead by the same method of Eric Tressler. But, when we get the roots, it is seen that two complex roots are conjugate and so form the factor $x^2-x+2$. Hence, the factorization of Veritas is obtained. I think with this way Veritas get the factorizati... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | martin | 13,496 | <p>To give a German-centric answer which would certianly apply to other countries as well: </p>
<p>The German Abitur is the right to study any subject at a university (Allgemeine Hochschulreife). As such the pupils need to be taught everything such that they need in order to be able to succeed. After all, they may wan... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Russell McMahon | 13,518 | <p>We are allowing people with less than even basic numeracy to leave school.<br>
We are allowing people to enter university with not only no knowledge of basic algebra or calculus but also with no realisation that basic algebra and calculus are essential tools when dealing with subjects such as history or social scien... |
3,043,699 | <p>I'm having difficulty understanding how below integral is evaluated : </p>
<p><span class="math-container">$$\int_0^1\frac{(1-y)^2}{2}+(1-y)y dy = \frac{-1(1-y)^3}{6}+\frac{y^2}{2}-\frac{y^3}{3}\bigg|^1_0$$</span></p>
<p>What are steps involved in this evaluation ?</p>
<p>For <span class="math-container">$\frac{(... | Community | -1 | <p>By the chain rule,</p>
<p><span class="math-container">$$((ax+b)^k)'=(ax+b)'k(ax+b)^{k-1}=ak(ax+b)^k$$</span></p>
<p>and from this,</p>
<p><span class="math-container">$$\int(ax+b)^kdx=\frac{(ax+b)^{k+1}}{a(k+1)}.$$</span></p>
<p>Try with <span class="math-container">$a=-1$</span>.</p>
|
2,534,369 | <p>I am trying to work my through the exercises in Spivak's <em>Calculus on Manifolds.</em> I am currently working on the exercises in Chapter 3 which deals with Integration. I am having trouble with the following question:</p>
<blockquote>
<p>Let:</p>
<p>\begin{equation}
f(x,y)=\begin{cases}
0, & \te... | RAUL PEREZ | 963,490 | <p>It is obvious that the inferior sums are zero, so we want to proof that the superior sums <span class="math-container">$U(f;P)$</span> can be taken less than any previously fixed <span class="math-container">$\epsilon$</span>. Let p prime and let <span class="math-container">$k=\max \{n \mid n(n+1)/2 +1\leq p)\}$</s... |
123,675 | <p>What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas... | Mike Shulman | 49 | <p>This is probably too advanced for a first course, but you might be interested in Ieke Moerdijk's monograph "Classifying Spaces and Classifying Topoi". Among other things, it uses topos theory to show what it is that the "classifying space" of a (non-groupoid) category classifies. IIRC this was an ingredient in the... |
2,008,437 | <p>Given four points in the plane, there exists a one-dimensional family of conics through these, often called a pencil of conics. The locus of the centers of symmetry for all of these conics is again a conic. What's the most elegant way of computing it?</p>
<p>I know I could choose five arbitrary elements from the pe... | Blue | 409 | <p>Here's a coordinate derivation that's computationally intense, although my copy of Mathematica was able to crunch the symbols fairly quickly.</p>
<hr>
<p>Let the points $P$, $Q$, $R$, $S$ be arranged such that segments $\overline{PR}$ and $\overline{QS}$ meet at the origin, and such that the coordinate axes bisect... |
1,085,164 | <p><strong>I am currently taking a course on Numerical PDE. The course covers the following topics listed below.</strong> </p>
<p><strong>Chapter 1: Solutions to Partial Dierential Equations:</strong></p>
<p><strong>Chapter 2: Introduction to Finite Elements:</strong></p>
| dustin | 78,317 | <p>A book I like, but I am probably biased since he was my professor is Dr. Jianke Yang's book <a href="http://rads.stackoverflow.com/amzn/click/0898717051" rel="nofollow">Nonlinear Waves in Integrable and Non-integrable systems</a></p>
<p>It covers:</p>
<ol>
<li>Derivation of non-linear waves</li>
<li>Integrable the... |
2,315,599 | <p>I am trying to evaluate the integral<br>
$\int (\frac{1}{z(\exp(z)-1)}dz$ $\mathbb{C}:|z|=1\}$</p>
<p>using Laurent theorem. $0$ is a singularity point and both denominators becomes $0$ when $z=0$. So it's hard to find a method to evaluate this using Residue theorem. Could someone please show me how to solve thi... | Johnq | 233,719 | <p>Soumee,</p>
<p>I believe I may have an answer that is founded in equations based off your idea from before.</p>
<p>Set up a reference frame with left denoting the negative direction and right being positive. Place the girl's house at the origin. Using the 1-D kinematics equations for position, let the following be... |
423,302 | <p>I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, <em>I will cheat</em>. Whenever I fail to have enough evidence, I will simply increase the sample size by continuing to flip the coin.</p>
<p>To be specific, I will flip the coin un... | not all wrong | 37,268 | <p><strong>Edited Note</strong>: From David Speyer's answer the below reaches the correct conclusion -- divergence in the expectation value -- though I haven't verified the intermediate observation about the actual asymptotics. (There was a bit of confusion at one point.) </p>
<hr>
<p>I ran 10000000 simulations (up t... |
423,302 | <p>I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, <em>I will cheat</em>. Whenever I fail to have enough evidence, I will simply increase the sample size by continuing to flip the coin.</p>
<p>To be specific, I will flip the coin un... | David E Speyer | 448 | <p>According to Blackwell and Freedman, <a href="http://projecteuclid.org/euclid.aoms/1177703292" rel="nofollow">A Remark on the Coin Tossing Game</a>, the expected time for the number of heads to get outside $n/2 \pm a \sqrt{n}$ is finite for $a<0.5$ and infinite for $a \geq 0.5$. (As Henry points out below, they u... |
2,036,278 | <p>I have a question about inverse images and containment. </p>
<p>Let $\;f: A \rightarrow B, D \subseteq A$, and $E \subseteq B$. Show that $\; f^{-1}(B-E) \subseteq A-f^{-1}(E)$. </p>
<p>I have started by letting $x \in f^{-1}(B-E)$ so $f(x) \in (B-E)$, but I am unsure where to go from here.</p>
| Olivier Oloa | 118,798 | <p>Setting
$$
S_N:=\sum_{l=2}^N{\frac{1}{l^m}}, \quad m \ge 3
$$ we have
$$
S_{N+1}-S_N=\frac{1}{l^m}>0,
$$
$\{S_N\}$ is an increasing sequence then observing that, for $m\ge3$,
$$
S_N=\sum_{l=2}^N{\frac{1}{l^m}}<\sum_{l=2}^N{\frac{1}{l(l+1)}}=\frac12-\frac1{N+1}<\frac12
$$ $\{S_N\}$ is then a bounded seque... |
4,263,836 | <blockquote>
<p>I would like to compute <span class="math-container">$\mathrm{Ext}^i_{\mathbb Z}(\mathbb Q, \mathbb Z/2\mathbb Z)$</span>.</p>
</blockquote>
<p>So from the definition I learned in my class, I need to find a free resolution of <span class="math-container">$\mathbb Q$</span> over <span class="math-contain... | Thomas Andrews | 7,933 | <p>If you must do it with a free resolution, consider the free group <span class="math-container">$\oplus_{i=1}^\infty \mathbb Z$</span> and morphism:</p>
<p><span class="math-container">$$\phi:(a_i)_{i=1}^\infty \mapsto \sum_{i=1}^\infty \frac{a_i}{i!}$$</span></p>
<p>Then the kernel has as a basis <span class="math-c... |
3,676,127 | <p>Is there a term in mathematics for the <strong>SET</strong> of all possible <span class="math-container">$n \times n$</span> zero-one matrices?</p>
<p>In other words, for a particular <span class="math-container">$n \in \mathbb{Z}^+$</span>, is there a term for the set of all possible <span class="math-container">$... | J.-E. Pin | 89,374 | <p>It is the set <span class="math-container">${\Bbb B}^{n \times n}$</span> of <span class="math-container">$n \times n$</span> Boolean matrices. You can extend the notation to <span class="math-container">${\Bbb B}^{n \times m}$</span>, the set of <span class="math-container">$n \times m$</span>-matrices with entries... |
1,842,093 | <p><a href="https://i.stack.imgur.com/A5U3D.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A5U3D.png" alt="enter image description here"></a></p>
<p>$ABCD$ forms a square. $CDE$ forms a triangle. Given $\measuredangle AED=15^{\circ}$ and $DE=CE$, prove $\triangle CDE$ is equilateral. </p>
<p>The q... | Nathaniel B | 204,043 | <p>Each possible length of $DE$ both is uniquely determined by and uniquely determines angle $\measuredangle CDE$. Likewise, $\measuredangle DEA$ is uniquely determined by and uniquely determines either one of the length $DE$ or $\measuredangle CDE$. This means that we can reverse the problem! Thus "Prove that if trian... |
2,976,057 | <p>I’m doing partial fractions and need to factorize the denominator. They are quadratic. However there are some that aren’t so easy to factorize and my first choice was to use the quadratic equation to find the roots however comparing my answer with the correct one the signs are different. Is the quadratic formula onl... | Joppy | 431,940 | <p>Suppose I have a mystery quadratic <span class="math-container">$p(x) = -2x^2 + x + 1$</span>, and want to factorise it using the quadratic formula. I can find the roots easily enough:</p>
<p><span class="math-container">$$ x = \frac{-1 \pm \sqrt{1 + 8}}{-4}$$</span>
and so the roots are <span class="math-container... |
663,218 | <p>Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ,\rangle$ of $\mathbb R^n$. Does not have to be the standard one, just some random inner product.</p>
<p>let $T^*$ be ... | Yiorgos S. Smyrlis | 57,021 | <p>No it does not hold for any inner product.</p>
<p>It is hard hard to show that any other inner product $\langle\cdot,\cdot\rangle_*$ can be represented as
$$
\langle x,y\rangle_\star=\langle x,Sy\rangle,
$$
where $S$ is a positive definite matrix. So
$$
\langle Tx,y\rangle_\star=\langle x,T^*Sy\rangle \ne \langle x... |
2,027,529 | <p>I want to show that $\pi_1(S^1\vee S^1) = \mathbb{Z}*\mathbb{Z}. $ I know that it follows from the Seifert-van Kampen theorem, but we haven't talked about that in class. </p>
<p>We are given the following hint:</p>
<blockquote>
<p>Define a group homomorphism $\mathbb{Z}*\mathbb{Z}\to\pi_1(S^1\vee S^1)$ via the t... | Ben Grossmann | 81,360 | <p>Present the group $\Bbb Z * \Bbb Z = \langle a,b \rangle$. It suffices to define our map on the generators $a,b$.</p>
<p>Let $L(S^1 \vee S^1)$ denote the group of loops over composition, and $\sim$ be the homotopy equivalence relation so that $\pi_1(X) = L(X)/\sim$. Let $f$ be a map that sends $a$ to a loop aroun... |
1,620,795 | <p>Find function f(x), where:
$$f(3)=3$$
$$f'(3)=3$$
$$f'(4)=4$$
$$f''(3) = \nexists$$ </p>
<p>How to find function like this in <strong>general?</strong> What steps should I do?</p>
| TZakrevskiy | 77,314 | <p>$$|x\cdot y| \le \sum_{i=1}^n |x_i\bar y_i|\le \sum_{i}\left((\sup_j|x_j||y_i|)\right)=\sup_j|x_j|\sum_i|y_i|=\|x\|_{\infty}\|y\|_1$$</p>
|
607,353 | <p>So I've recently been looking at the <a href="http://www.photographyblogger.net/stereographic-projection-photography-100-little-planets/" rel="nofollow">Tiny Planet</a> images.</p>
<p>I've been googling a few things to try and find out how images are converted from normal to a tiny planet. Some phone apps, as well ... | E.O. | 18,873 | <p>The process of creating the images you describe uses a kind of projection called <strong>stereographic projection</strong>. When you take your panorama photo, it "stores" the data on a sphere so to speak.</p>
<p>$\hspace{45mm}$<img src="https://i.stack.imgur.com/9Y3du.png" alt="enter image description here"></p>
<... |
1,455,098 | <p>I am reading the book Foundation of Machine Learning, and the author has many proofs for different theorems. Here is one part of a proof about Perceptron algorithm which I don't quite understand</p>
<p>Note: $w_t$ is a vector</p>
<p>Let
\begin{equation}
w_{t+1} = \begin{cases}
w_t + ny_t x_t & \text{if } y_t(... | Sriharsha Madala | 254,941 | <p>From my understanding of perceptron, in the training scenario, for every input, you change the weights in such a way to minimize the error (typically mean squared error). The delta weight vector is perpendicular to the current weight vector.</p>
<p>Hence $\|w_{t+1} \|^2 = \|w_{t} \|^2 + \|ny_tx_t \|^2 (\text{ if }... |
1,408,231 | <p>Definition </p>
<p>A function $f$ is <strong>convex</strong> on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have</p>
<p>$$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$</p>
<p>While reading the proof that if $f$ is convex on some open interval containing $a$, then $f_+'(a)$ ... | zhw. | 228,045 | <p>I'm going to suggest you think geometrically here (which includes drawing a picture): Suppose $A,B,C$ are points in the plane with $A$ to the left of $B$ and $B$ to the left of $C.$ The first inequality you cite is of the form $s(A,B) \le s(A,C),$ where $s$ denotes slope. How could this be true unless $B$ lies on or... |
1,712,933 | <p>I am slightly ashamed to be asking this, but I have been recently reflecting on changing variables in very simple problems. If I missed a question that already discusses this please point it out to me and I will delete this one.
Anyhow writing this will probably be a learning experience.</p>
<p>Directly from the Wi... | Ethan Bolker | 72,858 | <p>Good question, good answers.</p>
<p>As @EricS points out, you don't have to substitute in this particular case - you can do all the work with the original variable. But you can substitute. The advantage is that changing the name of the variable in a systematic way makes the shape of the problem and solution a littl... |
95,343 | <p>I am writing a code for a Lagrange multipliers problem with three variables.
My code looks like this:</p>
<pre><code> f[x_, y_, z_] := (m1 x + m2 y + m3 z)/Sqrt[m1^2 x + m2^2 y + m3^2 z]
g[x_, y_, z_] := x + y + z - 1
Minimize[{f[x, y, z], g[x, y, z] == 0,
m1 > 0, m2 > 0, m3 > 0, x > 0, y... | Community | -1 | <p>To confirm @belisarius comment:</p>
<pre><code>f[x_, y_, z_] := (m1 x + m2 y + m3 z)/Sqrt[m1^2 x + m2^2 y + m3^2 z]
g[x_, y_, z_] := x + y + z - 1
L = f[x, y, z] + λ g[x, y, z];
Solve[{Grad[L, {x, y, z}] == 0, g[x, y, z] == 0, m1 > 0, m2 > 0,
m3 > 0, x > 0, y > 0, z > 0}, {x, y, z, λ}]
{}
... |
3,439,833 | <p><strong>Problem:</strong>
Let <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R}$</span> be differentiable.
Suppose <span class="math-container">$f(0)=0$</span>, and <span class="math-container">$f'(x) < \frac{1}{2}$</span> for all <span class="math-container">$a$</span>. Show <span class="math-con... | Peter Szilas | 408,605 | <p><span class="math-container">$f(4)-f(0)=\displaystyle{\int_{0}^{4}}f'(x)dx< $</span></p>
<p><span class="math-container">$\displaystyle{\int_{0}^{4}}(1/2)dx =(1/2)4=2$</span>.</p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Mark Wildon | 7,709 | <p>Let <span class="math-container">$f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$</span> be a boolean function. Ordering <span class="math-container">$\mathbb{F}_2$</span> so that <span class="math-container">$0<1$</span>, we say that <span class="math-container">$f$</span> is <em>monotone</em> if <span class="math-... |
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Mike Earnest | 59,232 | <p>A ranked poset is called a <em>symmetric chain order</em>, or SCO, if it can partitioned into rank symmetric, saturated chains. Such posets are particularly nice, as they are rank-symmetric, unimodal, and have the strong Sperner property. If <span class="math-container">$P$</span> and <span class="math-container">$Q... |
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Joseph Van Name | 22,277 | <p>So in mathematics, one can iterate operation of mapping a group to its automorphism group transfinitely to obtain the automorphism tower of a group. And <a href="https://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1336588247" rel="nofollow noreferrer">the automorphism tower always terminates</a>. The next natu... |
244,875 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
s.t.~\lambda^TA\leq c^T
\end{align}</p>
<p>But in the constraint, I always get an equality (using what I learnt)
\begin{align}... | Keith Reyes | 49,970 | <p>Given </p>
<p>$\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}$</p>
<p>I think the dual of the problem would be</p>
<p>$\begin{align}
\max_{x} b^Tw \\
s.t.~A^Tw (*)c
\end{align}$</p>
<p>where $(*)$ actually depends on the restrictions of the variables in the primal problem. If the restriction is $x_i \geq 0... |
2,509,583 | <p>How is intermediate value theorem valid for $\sin x$ in $[0,\pi]$?</p>
<p>It has max value $1$ in the interval $[0,\pi]$ which doesn't lie between values given by $\sin0$ and $\sin\pi$.</p>
| gt6989b | 16,192 | <p><a href="https://en.wikipedia.org/wiki/Intermediate_value_theorem" rel="nofollow noreferrer">Intermediate Value Theorem</a> says that </p>
<blockquote>
<p>Let $f$ be a function, continuous on the interval, $[a, b]$. Then $f$ takes any value between $f(a)$ and $f(b)$ at some point within the interval.</p>
</blockq... |
3,603,926 | <p>I came across this elementary counting problem: A coffee shop has 4 different types of coffee. You can order your coffee in a small, medium, or large cup. You can also choose whether you want to add cream, sugar, or milk (any combination is possible, for example, you can choose to add all three). In how many ways ca... | Arthur | 15,500 | <p>Yup, that is the answer. You have four options for coffee, three options for size, two options for cream, two options for sugar, and two options for milk, and you have to pick one option for each of these five choices. This does indeed give <span class="math-container">$$4\cdot 3\cdot 2\cdot 2\cdot 2$$</span></p>
|
221,137 | <p>What's the difference between Fourier transformations and Fourier Series? </p>
<p>Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?</p>
| Shivam Kumar | 90,692 | <p>Fourier transform is used to transform periodic and non-periodic signals from time domain to frequency domain. It can also transform Fourier series into the frequency domain, as Fourier series is nothing but a simplified form of time domain periodic function.</p>
<h3>Fourier series</h3>
<ol>
<li><p>Periodic functi... |
2,948,673 | <p>I was thinking about the squeeze theorem here. We can denote the <span class="math-container">$\left[\frac{1}{|x|}\right] =n$</span>, and then try something like:</p>
<p><span class="math-container">$$x^2(1+2+3+...+(n-1)+(n-1)) \leq x^2 \frac{n(n+1)}{2} \leq x^2(1+2+3+...+(n-1)+(n+1))$$</span></p>
<p>But I don't k... | MPW | 113,214 | <p>Why not just substitute <span class="math-container">$x=1/n$</span> (noting that the limits from both sides are the same) and consider <span class="math-container">$$\lim\limits_{n\to\infty} \frac1{n^2}(1+2+\cdots+n)$$</span>
instead?</p>
|
1,221,368 | <p>An inheritance is divided between three people in the ratio $4:7:2$. If the least amount received is $\$\ 2300$ calculate how much the other two people received?</p>
| direprobs | 226,619 | <p>The answer is:</p>
<p>we have 13 parts in the total amount: 4 + 7 + 2 = 13 parts
The lowest person receives 2 parts that is 2300 and so 2300/2= 1150 for each part. Thus the total amount is (1150 * 13) = 14950</p>
<p>Therefore:</p>
<blockquote>
<p>2/13 * 14950 = 2300</p>
<p>4/13 * 14950 = 4600</p>
<p>7/13 * 14950 = 8... |
103,358 | <p>I am trying to illustrate some simple ideas with exponents. I can manually express something like $5^4$ as $5 \cdot 5 \cdot 5 \cdot 5$, but wondered how to get <em>Mathematica</em> to do that for me.</p>
<p>I found the example below in the documentation, but can't figure out how to "massage" it to work with a num... | SC_L | 30,459 | <p>I got the same problem and solved it by doing this. Just go to the <strong>Preferences</strong>-><strong>Internet Connectivity</strong> and turn off <strong>Allow the Wolfram System to access the Internet</strong>, then relaunch Mathematica.</p>
|
103,358 | <p>I am trying to illustrate some simple ideas with exponents. I can manually express something like $5^4$ as $5 \cdot 5 \cdot 5 \cdot 5$, but wondered how to get <em>Mathematica</em> to do that for me.</p>
<p>I found the example below in the documentation, but can't figure out how to "massage" it to work with a num... | level1807 | 5,784 | <p>This annoying bug has finally been fixed in 11.0</p>
|
359,816 | <p>Let $(f_n)$ and $(g_n)$ be sequences of nonnegative function in $L^1(\mathbb R)$, for which
$$
f_n \to 0, \\
g_n \to 0,
$$
almost everywhere.
Show
$$
\int_A \frac{2f_n g_n}{1+f_n^2+g_n^2} \to 0,
$$
when $A \subset \mathbb R$ is a set of finite measure.</p>
<p>Define
$$
h_n = \frac{2f_n g_n}{1+f_n^2+g_n^2}.
$$
I kno... | Umberto P. | 67,536 | <p>You have $0 \le h_n \le g_n$ since $\dfrac{2f_n}{1 + f_n^2} \le 1$. Thus $h_n \to 0$. You also have $0 \le h_n \le 1$ since $\dfrac{2f_n g_n}{1 + f_n^2 + g_n^2} \le 1$. Since $A$ has finite measure, this provides a majorant of the sequence.</p>
|
121,760 | <p>I want to find the minimal polynomial (over $\mathbb{Q}$) of: $k:=\sqrt[3]{7-\sqrt{2}}$.</p>
<p>With simple 'tricks' I got that: $P=(x^3-7)^2+2$ is a polynomial such that $P(k)=0$.</p>
<p>But I don't know if, or how to prove that $P$ is the minimal polynomial. How can I prove this/find the minimal polynomial ? </p... | Zarrax | 3,035 | <p>Let $\alpha = \sqrt[3]{7 - \sqrt{2}}$ and $P(x) = x^6 - 14x^3 + 47$. Since $P(\alpha) = 0$, in order to show $P$ is irreducible it suffices to show $[Q(\alpha):Q] = 6$. Since $7 - \alpha^3 = \sqrt{2}$, you have that $\sqrt{2} \in Q(\alpha)$ and thus
$$[Q(\alpha):Q] =[Q(\alpha):Q(\sqrt{2})][Q(\sqrt{2}):Q] = 2[Q(\alp... |
1,425,961 | <p>I am about to take a course in undergraduate Algebra which is using Artin's Algebra. I have been studying from it and haven't really appreciated the combination of text and problems (granted, I haven't gone too deep into the text). </p>
<p>Now, this is the text for the class, so I would expect a nice compliment f... | Community | -1 | <p>I have and am still frequently using both books and I have to say the main thing they have going is that :</p>
<ul>
<li>D&F has great exercises.</li>
<li>Artin is friendlier to the audience.</li>
</ul>
<p>My usual advice is that one learns from Artin and then does the exercises in D&F, but overall they are... |
483,602 | <p>$$
\large\sqrt{8+\sqrt{5}-\sqrt{6-2\sqrt{5}}}
$$</p>
<p>I am asked to write the following as a ratio of two integers (i.e.,) show that this value is rational
(Do not use calculator)</p>
<p>What I tried doing was putting everything to the exponent of $.5$ and then was unsure how to proceed </p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> $6-2\sqrt{5}=(\sqrt{5}-1)^2$. </p>
|
1,808,019 | <p>I'm trying to better understand how the following group action looks like on a torus. The group $\mathbb{R}$ acts on the torus $C\times C=\{(e^{ix},e^{iy}):x,y\in \mathbb{R}\}$ ($C$ being the unit circle in $\mathbb{C}$) via $$t\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+t)},e^{iy})$$</p>
<p>I am able (I think) to visuali... | Surb | 154,545 | <p><strong>Just a complement : Proof using combinatorial argument</strong></p>
<p>Let $X$ a set with $x$ elements, $Y$ set with $y$ elements s.t. $X\cap Y=\emptyset$ and $N$ a set with $n$ element.
$$(x+y)^n=\#\{f:N\to X\cup Y\mid f\text{ is a function}\}.$$</p>
<p>An other way to count such a function is the followi... |
70,158 | <p>This question is related to <a href="https://mathoverflow.net/questions/52825/coloring-mathbb-zk"> this one </a> but feels more Ramsey-type, so perhaps it is easier. Let $S$ be a finite set, $|S|=k$. Suppose we color all subsets of $S$ in $1000$ colors. What is the maximal (in terms of $k$) guaranteed length $l=l(k)... | Tony Huynh | 2,233 | <p>Here is a proof of a very weak upper bound for $l(k)$. Consider the colouring of $2^S$ where each set is coloured by its size (mod 1000). A good monochomatic sequence must consist of sets of the same size. Thus we obtain $l(k) \leq \binom{k}{k/2}$. </p>
<p>However, we can be a bit smarter. Instead of colouring... |
3,669,961 | <p>Let <span class="math-container">$f:[a,b] \longrightarrow \mathbb{R}$</span> be a integrable function. I know if <span class="math-container">$ f> 0 $</span> then
<span class="math-container">$$\int_{a}^{b} f(x)\; dx >0.$$</span> The converse is true? That is, if <span class="math-container">$$\int_{a}^{b} f(x... | José Carlos Santos | 446,262 | <p>What about <span class="math-container">$\displaystyle\int_{-1}^3x\,\mathrm dx$</span>, which happens to be equal to <span class="math-container">$4$</span>?</p>
|
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Community | -1 | <p>For number 1, when a statement in the representation theory of finite groups is true over <span class="math-container">$\mathbb{C}$</span>, I think it should be true for any algebraically closed field of characteristic zero. That's because the development of the representation theory of finite groups proceeds the sa... |
191,262 | <p>I have to fit a Cos^2 function to data I measured. The function is <span class="math-container">$a \cos^2(\frac{bx\pi}{180}+c)+d $</span> and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the first one worked. The other three only yielded not usable fits. I am pretty new to Ma... | JimB | 19,758 | <p>The two answers so far (@MassDefect and @m0nhawk) are exactly what you want to do: have good initial guesses (especially for anything dealing with sines and cosines).</p>
<p>If you are fitting many sets of data to the same model, then automating the initial values is recommended. For your example, you could use:<... |
1,005,921 | <p>I was just wondering if anyone could shed more light on specific topics in applied mathematics or other skills (programming, etc) commonly used in the following industries: oil and gas, aviation/aeronautics. In particular, I'm interested in how mathematical modelling could be applied in these industries (from the pe... | 5xum | 112,884 | <p>I don't know about digits of $\pi$, but if you have a truly random generator, then "lags" of certain digits will not correct themselves.</p>
<p>By that I mean the following:</p>
<p>Say you have a random generator that returns integers from $0$ to $9$, and say that in the last one million tries, the generator gener... |
2,744,172 | <p>(Note: This question is tangentially related to this <a href="https://math.stackexchange.com/q/3609304">later one</a>.)</p>
<p>Let <span class="math-container">$$\sigma(x) = \sum_{d \mid x}{d}$$</span> denote the <em>sum of divisors</em> of <span class="math-container">$x \in \mathbb{N}$</span>, where <span class=... | Jose Arnaldo Bebita Dris | 28,816 | <p>(This is not an answer - just some comments that are too long to fit in the appropriate section.)</p>
<p>Note that the inequality <span class="math-container">$m < k$</span> is equivalent to the assertion "<span class="math-container">$k$</span> is not an odd almost perfect number", where <span class="math-conta... |
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